ukaea · timothy-nunn · May 23, 2024 · May 23, 2024 · May 23, 2024 · May 23, 2024
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+# Fast Alpha Pressure Contribution
+
+The pressure contribution from the fast alpha particles can be controlled using switch `ifalphap`. 
+There are two options 1[^1] and 2[^2]:
+
+$$\begin{aligned}
+\frac{\beta_{\alpha}}{\beta_{th}} & = 0.29 \, \left( \langle T_{10} \rangle -
+  0.37 \right) \, \left( \frac{n_{DT}}{n_e} \right)^2
+\hspace{20mm} \mbox{if alphap = 0} \\
+\frac{\beta_{\alpha}}{\beta_{th}} & = 0.26 \, \left( \langle T_{10} \rangle -
+  0.65 \right)^{0.5} \, \left( \frac{n_{DT}}{n_e} \right)^2
+\hspace{16mm} \mbox{if alphap = 1 (default)}
+\end{aligned}$$
+
+The latter model is a better estimate at higher temperatures.
+
+[^1]: T. C. Hender et al., 'Physics Assessment for the European Reactor Study',
+
+[^2]: H. Lux, R. Kemp, D.J. Ward, M. Sertoli, 'Impurity radiation in DEMO 
+systems modelling', Fus. Eng.  | Des. **101**, 42-51 (2015)
@@ -0,0 +1,46 @@
+# Beta Limit
+
+The plasma beta limit[^1] is given by 
+
+$$\begin{aligned}
+\beta < g \, \frac{I(\mbox{MA})}{a(\mbox{m}) \, B_0(\mbox{T})}
+\end{aligned}$$
+
+where $B_0$ is the axial vacuum toroidal field. The beta
+coefficient $g$ is set using input parameter `dnbeta`. To apply the beta limit, 
+constraint equation 24 should be turned on with iteration variable 36
+(`fbetatry`). 
+
+By default, $\beta$ is defined with respect to the total equilibrium B-field [^2]. 
+
+| `iculbl` | Description |
+| :-: | - |
+| 0 (default) | Apply the $\beta$ limit to the total plasma beta (including the contribution from fast ions) |
+| 1 | Apply the $\beta$ limit to only the thermal component of beta |
+| 2 | Apply the $\beta$ limit to only the thermal plus neutral beam contributions to beta |
+| 3 | Apply the $\beta$ limit to the total beta (including the contribution from fast ions), calculated using only the toroidal field |
+
+### Scaling of beta $g$ coefficient
+
+Switch `gtscale` determines how the beta $g$ coefficient `dnbeta` should 
+be calculated, using the inverse aspect ratio $\epsilon = a/R$.
+
+| `gtscale` | Description |
+| :-: | - |
+| 0 | `dnbeta` is an input. |
+| 1 | $g=2.7(1+5\epsilon^{3.5})$ (which gives g = 3.0 for aspect ratio = 3) |
+| 2 | $g=3.12+3.5\epsilon^{1.7}$ (based on Menard et al. "Fusion Nuclear Science Facilities and Pilot Plants Based on the Spherical Tokamak", Nucl. Fusion, 2016, 44)  |
+
+!!! Note 
+    `gtscale` is over-ridden if `iprofile` = 1.
+
+### Limiting $\epsilon\beta_p$
+
+To apply a limit to the value of $\epsilon\beta_p$, where $\epsilon = a/R$ is
+the inverse aspect ratio and $\beta_p$ is the poloidal $\beta$, constraint equation no. 6 should be 
+turned on with iteration variable no. 8 (`fbeta`). The limiting value of $\epsilon\beta_p$ 
+is be set using input parameter `epbetmax`.
+
+[^1]: N.A. Uckan and ITER Physics Group, 'ITER Physics Design Guidelines: 1989',
+
+[^2]: D.J. Ward, 'PROCESS Fast Alpha Pressure', Work File Note F/PL/PJK/PROCESS/CODE/050
@@ -0,0 +1,155 @@
+# Confinement Time Scaling Laws
+
+The energy confinement time $\tau_E$ is calculated using one of a choice of empirical scalings. ($\tau_E$ is defined below.)
+
+Many energy confinement time scaling laws are available within PROCESS, for
+tokamaks, RFPs and stellarators. These are calculated in routine `pcond`. The 
+value of `isc` determines which of the scalings is used in the plasma energy 
+balance calculation. The table below summarises the available scaling laws. The 
+most commonly used is the so-called IPB98(y,2) scaling.       
+
+| `isc` | scaling law | reference |
+| :-: | - | - |
+| 1 | Neo-Alcator (ohmic) | [^1] | 
+| 2 | Mirnov (H-mode) | [^1] |
+| 3 | Merezhkin-Muhkovatov (L-mode) | [^1] |
+| 4 | Shimomura (H-mode) | JAERI-M 87-080 (1987) |
+| 5 | Kaye-Goldston (L-mode) | Nuclear Fusion **25** (1985) p.65 |
+| 6 | ITER 89-P (L-mode) | Nuclear Fusion **30** (1990) p.1999 |
+| 7 | ITER 89-O (L-mode) | [^2] |
+| 8 | Rebut-Lallia (L-mode) | Plasma Physics and Controlled Nuclear Fusion Research **2** (1987) p. 187 |
+| 9  | Goldston (L-mode)| Plas.\ Phys.\ Controlled Fusion **26** (1984) p.87 |
+| 10 | T10 (L-mode) | [^2] |
+| 11 | JAERI-88 (L-mode) | JAERI-M 88-068 (1988) |
+| 12 | Kaye-Big Complex (L-mode) | Phys.\ Fluids B **2** (1990) p.2926 |
+| 13 | ITER H90-P (H-mode) |  |
+| 14 | ITER Mix (minimum of 6 and 7) |  |
+| 15 | Riedel (L-mode) |  |
+| 16 | Christiansen et al. (L-mode) | JET Report JET-P (1991) 03 |
+| 17 | Lackner-Gottardi (L-mode) | Nuclear Fusion **30** (1990) p.767  |
+| 18 | Neo-Kaye (L-mode) | [^2] |
+| 19 | Riedel (H-mode) |  |
+| 20 | ITER H90-P (amended) | Nuclear Fusion **32** (1992) p.318 |
+| 21 | Large Helical Device (stellarator) | Nuclear Fusion **30** (1990) |
+| 22 | Gyro-reduced Bohm (stellarator) | Bull. Am. Phys. Society, **34** (1989) p.1964 |
+| 23 | Lackner-Gottardi (stellarator) | Nuclear Fusion **30** (1990) p.767 |
+| 24 | ITER-93H (H-mode) | PPCF, Proc. 15th Int. Conf.Seville, 1994 IAEA-CN-60/E-P-3 |
+| 25 | TITAN (RFP) | TITAN RFP Fusion Reactor Study, Scoping Phase Report, UCLA-PPG-1100, page 5--9, Jan 1987 |
+| 26 | ITER H-97P ELM-free (H-mode) | J. G. Cordey et al., EPS Berchtesgaden, 1997 |
+| 27 | ITER H-97P ELMy (H-mode) | J. G. Cordey et al., EPS Berchtesgaden, 1997 |
+| 28 | ITER-96P (= ITER97-L) (L-mode) | Nuclear Fusion **37** (1997) p.1303 |
+| 29 | Valovic modified ELMy (H-mode) | |
+| 30 | Kaye PPPL April 98 (L-mode) |  |
+| 31 | ITERH-PB98P(y) (H-mode) | |
+| 32 | IPB98(y) (H-mode)   | Nuclear Fusion **39** (1999) p.2175, Table 5,  |
+| 33 | IPB98(y,1) (H-mode) | Nuclear Fusion **39** (1999) p.2175, Table 5, full data |
+| 34 | IPB98(y,2) (H-mode) | Nuclear Fusion **39** (1999) p.2175, Table 5, NBI only |
+| 35 | IPB98(y,3) (H-mode) | Nuclear Fusion **39** (1999) p.2175, Table 5, NBI only, no C-Mod |
+| 36 | IPB98(y,4) (H-mode) | Nuclear Fusion **39** (1999) p.2175, Table 5, NBI only ITER like |
+| 37 | ISS95 (stellarator) | Nuclear Fusion **36** (1996) p.1063  |
+| 38 | ISS04 (stellarator) | Nuclear Fusion **45** (2005) p.1684  |
+| 39 | DS03 (H-mode) | Plasma Phys. Control. Fusion **50** (2008) 043001, equation 4.13  |
+| 40 | Non-power law (H-mode) | A. Murari et al 2015 Nucl. Fusion 55 073009, Table 4.  |
+| 41 | Petty 2008 (H-mode) |  C.C. Petty 2008 Phys. Plasmas **15** 080501, equation 36 |
+| 42 | Lang 2012 (H-mode) | P.T. Lang et al. 2012 IAEA conference proceeding EX/P4-01 |
+| 43 | Hubbard 2017 -- nominal (I-mode) | A.E. Hubbard et al. 2017, Nuclear Fusion **57** 126039 |
+| 44 | Hubbard 2017 -- lower (I-mode) | A.E. Hubbard et al. 2017, Nuclear Fusion **57** 126039 |
+| 45 | Hubbard 2017 -- upper (I-mode) | A.E. Hubbard et al. 2017, Nuclear Fusion **57** 126039 |
+| 46 | NSTX (H-mode; spherical tokamak) | J. Menard 2019, Phil. Trans. R. Soc. A 377:201704401 |
+| 47 | NSTX-Petty08 Hybrid (H-mode) | J. Menard 2019, Phil. Trans. R. Soc. A 377:201704401 |
+| 48 | NSTX gyro-Bohm (Buxton) (H-mode; spherical tokamak) | P. Buxton et al. 2019 Plasma Phys. Control. Fusion 61 035006 |
+| 49 | Use input `tauee_in` |  |
+| 50 | ITPA20 (H-mode) | G. Verdoolaege et al 2021 Nucl. Fusion 61 076006 |
+
+### Effect of radiation on energy confinement
+
+Published confinement scalings are all based on low radiation pulses. A power
+plant will certainly be a high radiation machine --- both in the core, due to
+bremsstrahlung and synchrotron radiation, and in the edge due to impurity
+seeding. The scaling data do not predict this radiation --- that needs to be
+done by the radiation model. However, if the transport is very "stiff", as
+predicted by some models, then the additional radiation causes an almost equal
+drop in power transported by ions and electrons, leaving the confinement
+nearly unchanged.
+
+To allow for these uncertainties, three options are available, using the switch
+`iradloss`. In each case, the particle transport loss power `pscaling` is
+derived directly from the energy confinement scaling law.
+
+`iradloss = 0` -- Total power lost is scaling power plus radiation:
+
+`pscaling + pradpv = falpha*palppv + pchargepv + pohmpv + pinjmw/vol`
+
+
+`iradloss = 1` -- Total power lost is scaling power plus radiation from a region defined as the "core":
+
+`pscaling + pcoreradpv = falpha*palppv + pchargepv + pohmpv + pinjmw/vol`
+
+`iradloss = 2` -- Total power lost is scaling power only, with no additional 
+allowance for radiation. This is not recommended for power plant models.
+
+`pscaling = falpha*palppv + pchargepv + pohmpv + pinjmw/vol`
+
+## L-H Power Threshold Scalings
+
+Transitions from a standard confinement mode (L-mode) to an improved
+confinement regime (H-mode), called L-H transitions, are observed in most
+tokamaks. A range of scaling laws are available that provide estimates of the
+heating power required to initiate these transitions, via extrapolations
+from present-day devices. PROCESS calculates these power threshold values
+for the scaling laws listed in the table below, in routine `pthresh`.
+
+For an H-mode plasma, use input parameter `ilhthresh` to
+select the scaling to use, and turn on constraint equation no. 15 with
+iteration variable no. 103 (`flhthresh`). By default, this will ensure
+that the power reaching the divertor is at least equal to the threshold power
+calculated for the chosen scaling, which is a necessary condition for
+H-mode. 
+
+For an L-mode plasma, use input parameter `ilhthresh` to
+select the scaling to use, and turn on constraint equation no. 15 with 
+iteration variable no. 103 (`flhthresh`). Set lower and upper bounds for 
+the f-value `boundl(103) = 0.001` and `boundu(103) = 1.0` 
+to ensure that the power does not exceed the calculated threshold, 
+and therefore the machine remains in L-mode.
+
+
+| `ilhthresh` | Name | Reference |
+| :-: | - | - |
+| 1 | ITER 1996 nominal | ITER Physics Design Description Document |
+| 2 | ITER 1996 upper bound | D. Boucher, p.2-2 |
+| 3 | ITER 1996 lower bound | 
+| 4 | ITER 1997 excluding elongation | J. A. Snipes, ITER H-mode Threshold Database |
+| 5 | ITER 1997 including elongation |  Working Group, Controlled Fusion and Plasma Physics, 24th EPS conference, Berchtesgaden, June 1997, vol.21A, part III, p.961 |
+| 6 | Martin 2008 nominal | Martin et al, 11th IAEA Tech. Meeting |
+| 7 | Martin 2008 95% upper bound |  H-mode Physics and Transport Barriers, Journal |
+| 8 | Martin 2008 95% lower bound |  of Physics: Conference Series **123**, 2008 |
+| 9 | Snipes 2000 nominal | J. A. Snipes and the International H-mode |
+| 10| Snipes 2000 upper bound | Threshold Database Working Group |
+| 11| Snipes 2000 lower bound |  2000, Plasma Phys. Control. Fusion, 42, A299 |
+| 12| Snipes 2000 (closed divertor): nominal | 
+| 13| Snipes 2000 (closed divertor): upper bound | 
+| 14| Snipes 2000 (closed divertor): lower bound | 
+| 15| Hubbard 2012 L-I threshold scaling: nominal | [Hubbard et al. (2012; Nucl. Fusion 52 114009)](https://iopscience.iop.org/article/10.1088/0029-5515/52/11/114009) |
+| 16| Hubbard 2012 L-I threshold scaling: lower bound | [Hubbard et al. (2012; Nucl. Fusion 52 114009)](https://iopscience.iop.org/article/10.1088/0029-5515/52/11/114009 |
+| 17| Hubbard 2012 L-I threshold scaling: upper bound | [Hubbard et al. (2012; Nucl. Fusion 52 114009)](https://iopscience.iop.org/article/10.1088/0029-5515/52/11/114009 |
+| 18| Hubbard 2017 L-I threshold scaling | [Hubbard et al. (2017; Nucl. Fusion 57 126039)](https://iopscience.iop.org/article/10.1088/1741-4326/aa8570) |
+| 19 | Martin 2008 aspect ratio corrected nominal | Martin et al (2008; J Phys Conf, 123, 012033) |
+| 20 | Martin 2008 aspect ratio corrected 95% upper bound | [Takizuka et al. (2004; Plasma Phys. Contol. Fusion, 46, A227)](https://iopscience.iop.org/article/10.1088/0741-3335/46/5A/024)  |
+| 21 | Martin 2008 aspect ratio corrected 95% lower bound |  
+
+## Ignition
+
+Switch `ignite` can be used to denote whether the plasma is ignited, i.e. fully self-sustaining 
+without the need for any injected auxiliary power during the burn. If `ignite` = 1, the calculated 
+injected power does not contribute to the plasma power balance, although the cost of the auxiliary 
+power system is taken into account (the system is then assumed to be required to provide heating 
+and/or current drive during the plasma start-up phase only). If `ignite` = 0, the plasma is not 
+ignited, and the auxiliary power is taken into account in the plasma power balance during the burn 
+phase. An ignited plasma will be difficult to control and is unlikely to be practical. This 
+option is not recommended.
+
+[^1]: T. C. Hender et al., 'Physics Assessment for the European Reactor Study',
+AEA Fusion Report AEA FUS 172 (1992)
+[^2]: N.A. Uckan and ITER Physics Group, 'ITER Physics Design Guidelines: 1989',
+ITER Documentation Series, No. 10, IAEA/ITER/DS/10 (1990)
@@ -0,0 +1,107 @@
+# Plasma Current Scaling Laws
+
+A number of plasma current scaling laws are available in PROCESS $[^1]. These are calculated in 
+routine `culcur`, which is called by `physics`. The safety factor $q_{95}$ required to prevent 
+disruptive MHD instabilities dictates the plasma current Ip:
+
+$$\begin{aligned}
+I_p = \frac{2\pi}{\mu_0} B_t \frac{a^2 f_q}{Rq_{95}}
+\end{aligned}$$
+
+The factor $f_q$ makes allowance for toroidal effects and plasma shaping (elongation and 
+triangularity). Several formulae for this factor are available [2,3] depending on the value of 
+the switch `icurr`, as follows:
+
+| `icurr` | Description |
+| :-: | - |
+| 1 | Peng analytic fit | 
+| 2 | Peng double null divertor scaling (ST)[^4] | 
+| 3 | Simple ITER scaling | 
+| 4 | Revised ITER scaling[^5]  $f_q = \frac{1.17-0.65\epsilon}{2(1-\epsilon^2)^2} (1 + \kappa_{95}^2 (1+2\delta_{95}^2 - 1.2\delta_{95}^3) )$| 
+| 5 | Todd empirical scaling, I | 
+| 6 | Todd empirical scaling, II | 
+| 7 | Connor-Hastie model | 
+| 8 | Sauter model, allows negative $\delta$ | 
+| 9 | Scaling for spherical tokamaks, based on a fit to a family of equilibria derived by Fiesta: $f_q = 0.538 (1 + 2.44\epsilon^{2.736}) \kappa^{2.154} \delta^{0.06}$|
+
+## Plasma Current Profile Consistency
+
+A limited degree of self-consistency between the plasma current profile and other parameters [^6] can be 
+enforced by setting switch `iprofile = 1`. This sets the current 
+profile peaking factor $\alpha_J$ (`alphaj`) and the normalised internal inductance $l_i$ (`rli`) using the 
+safety factor on axis `q0` and the cylindrical safety factor $q*$ (`qstar`):   
+
+$$\begin{aligned}
+\alpha_J = \frac{q*}{q_0} - 1
+\end{aligned}$$
+
+$$\begin{aligned}
+l_i = ln(1.65+0.89\alpha_J)
+\end{aligned}$$
+
+The beta $g$ coefficient `dnbeta` also scales with $l_i$, as described above.
+
+It is recommended that current scaling law `icurr = 4` is used if `iprofile = 1`. 
+Switch `gtscale` is over-ridden if `iprofile = 1`.
+
+## Bootstrap, Diamagnetic and Pfirsch-Schlüter Current Scalings
+
+The fraction of the plasma current provided by the bootstrap effect
+can be either input into the code directly, or calculated using one of four
+methods, as summarised here. Note that methods `ibss = 1-3` do not take into account the 
+existence of pedestals, whereas the Sauter et al. scaling 
+(`ibss = 4`) allows general profiles to be used. 
+
+| `ibss` | Description |
+| :-: | - |
+| 1 | ITER scaling -- To use the ITER scaling method for the bootstrap current fraction.  Set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$). This method is valid at high aspect ratio only.
+| 2 | General scaling -- To use a more general scaling method, set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$).
+| 3 | Numerically fitted scaling [^7] -- To use a numerically fitted scaling method, valid for all aspect ratios, set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$).
+| 4 | Sauter, Angioni and Lin-Liu scaling [^8] [^9] -- Set `bscfmax` to the maximum required bootstrap current fraction ($\leq 1$).
+
+!!! Note "Fixed Bootstrap Current"
+    Direct input -- To input the bootstrap current fraction directly, set `bscfmax` 
+    to $(-1)$ times the required value (e.g. -0.73 sets the bootstrap faction to 0.73).
+
+The diamagnetic current fraction $f_{dia}$ is strongly related to $\beta$ and is typically small,
+hence it is usually neglected.  For high $\beta$ plasmas, such as those at tight
+aspect ratio, it should be included and two scalings are offered.  If the diamagnetic
+current is expected to be above one per cent of the plasma current, a warning
+is issued to calculate it.
+
+`idia = 0` Diamagnetic current fraction is zero.
+
+`idia = 1` Diamagnetic current fraction is calculated using a fit to spherical tokamak calculations by Tim Hender:
+
+$$f_{dia} = \frac{\beta}{2.8}$$
+
+`idia = 2` Diamagnetic current fraction is calculated using a SCENE fit for all aspect ratios:
+
+$$f_{dia} = 0.414 \space \beta \space (\frac{0.1 q_{95}}{q_0} + 0.44)$$
+
+A similar scaling is available for the Pfirsch-Schlüter current fraction $f_{PS}$.  This is
+typically smaller than the diamagnetic current, but is negative.
+
+`ips = 0` Pfirsch-Schlüter current fraction is set to zero.
+
+`ips = 1` Pfirsch-Schlüter current fraction is calculated using a SCENE fit for all aspect ratios:
+
+$$ f_{PS} = -0.09 \beta $$
+
+There is no ability to input the diamagnetic and Pfirsch-Schlüter current
+directly.  In this case, it is recommended to turn off these two scalings 
+and to use the method of fixing the bootstrap current fraction.
+
+[^1]: D.J. Ward, 'PROCESS Fast Alpha Pressure', Work File Note F/PL/PJK/PROCESS/CODE/050
+[^2]:  Albajar, Nuclear Fusion **41** (2001) 665
+[^3]: M. Kovari, R. Kemp, H. Lux, P. Knight, J. Morris, D.J. Ward, '“PROCESS”: A systems code for fusion power plants—Part 1: Physics' Fusion Engineering and Design 89 (2014) 3054–3069
+[^4]: J.D. Galambos, 'STAR Code : Spherical Tokamak Analysis and Reactor Code',
+Unpublished internal Oak Ridge document.
+[^5]: W.M. Nevins, 'Summary Report: ITER Specialists' Meeting on Heating and
+Current Drive', ITER-TN-PH-8-4, 13--17 June 1988, Garching, FRG
+[^6]: Y. Sakamoto, 'Recent progress in vertical stability analysis in JA',
+Task meeting EU-JA #16, Fusion for Energy, Garching, 24--25 June 2014
+
+[^7]: H.R. Wilson, Nuclear Fusion **32** (1992) 257
+[^8]: O. Sauter, C. Angioni and Y.R. Lin-Liu, Physics of Plasmas **6** (1999) 2834 
+[^9]: O. Sauter, C. Angioni and Y.R. Lin-Liu, Physics of Plasmas **9** (2002) 5140
@@ -0,0 +1,19 @@
+# Density Limit
+
+Several density limit models[^1] are available in PROCESS. These are
+calculated in routine `culdlm`, which is called by `physics`. To enforce any of 
+these limits, turn on constraint equation no. 5 with iteration variable no. 9 
+(`fdene`). In addition, switch `idensl` must be set to the relevant value, as 
+follows:
+
+| `idensl` | Description |
+| :-: | - |
+| 1 | ASDEX model |
+| 2 | Borrass model for ITER, I |
+| 3 | Borrass model for ITER, II |
+| 4 | JET edge radiation model |
+| 5 | JET simplified model |
+| 6 | Hugill-Murakami $M.q$ model |
+| 7 | Greenwald model: $n_G=10^{14} \frac{I_p}{\pi a^2}$ where the units are m and ampere. For the Greenwald model the limit applies to the line-averaged electron density, not the volume-averaged density. |
+
+[^1]: T. C. Hender et al., 'Physics Assessment for the European Reactor Study',